| Lie group | |
|---|---|
| Name | Lie group |
| Field | Mathematics; applications in Quantum physics |
| Introduced | 19th century |
| Introduced by | Sophus Lie |
| Main concepts | Lie algebra, group theory, differential geometry |
Lie group
A Lie group is a mathematical structure combining the algebraic notion of a group with the smooth structure of a differentiable manifold. Lie groups provide the natural language for continuous symmetries in Quantum mechanics and Quantum field theory, underpinning conservation laws via Noether's theorem and determining the classification of particles by their transformation properties. Their representations and unitary realizations are central to the formulation of quantum systems, gauge theories, and the standard model of particle physics.
A Lie group G is a set equipped with a group law (multiplication and inversion) that are smooth maps on a differentiable manifold. Classic examples include the matrix groups such as GL(n, R), SL(n, C), SO(n), and SU(n), which are both matrix groups and smooth manifolds. Fundamental properties include connectedness, compactness, and simple versus semisimple structure; these properties influence physical uses such as selection rules and spectrum degeneracies. Topological invariants like the fundamental group π1(G) and higher homotopy groups affect global aspects of quantum states (for example, projective representations and spin structures). Foundational results include the existence of a corresponding Lie algebra and the exponential map relating algebra elements to group elements.
In quantum theory continuous symmetries described by Lie groups yield conserved observables through Noether's theorem and the correspondence between continuous transformations and self-adjoint generators. Spatial symmetries such as rotation symmetry are modeled by SO(3) and its double cover SU(2), explaining spin and the distinction between bosons and fermions. Internal symmetries in the Standard Model arise from product groups like SU(3), SU(2), and U(1), governing color, weak isospin, and hypercharge. Global and gauge symmetries are expressed by Lie groups and Lie algebras; the associated conserved currents and charges play a central role in scattering theory and selection rules used in experiments at institutions such as CERN and Fermilab.
A representation of a Lie group G on a vector space V implements group elements as linear transformations. In quantum physics the physically relevant representations are unitary representations on Hilbert spaces, ensuring probability conservation. Key theorems such as the Peter–Weyl theorem and the Stone–von Neumann theorem constrain possible representations for compact and Heisenberg-type groups respectively. Classification of irreducible unitary representations for non-compact groups (e.g., SL(2, C) relevant to relativistic spin) is more subtle and employs methods from harmonic analysis and the representation theory developed by mathematicians like Élie Cartan, Harish-Chandra, and George Mackey. Practical consequences include selection rules, multiplet structure, and the construction of state spaces for atomic, nuclear, and particle systems.
The Lie algebra g of a Lie group G is the tangent space at the identity equipped with the Lie bracket, encoding infinitesimal generators of continuous transformations. In quantum mechanics these generators correspond to Hermitian operators (via multiplication by i) such as angular momentum operators J_i satisfying commutation relations of the so(3) algebra. Structure constants, Cartan subalgebras, roots, and weight lattices determine representation theory and spectrum multiplicities. Exponentiation maps Lie algebra elements to group elements through the exponential map, and Baker–Campbell–Hausdorff formulas handle composition of infinitesimal transformations, important in perturbation theory and the derivation of effective Hamiltonians.
Lie groups organize conserved quantities and classification of states in atomic, molecular, and nuclear physics (e.g., SO(4) in the Kepler problem, SU(2), SU(3) flavor symmetry). In quantum field theory gauge groups like SU(3)_color, SU(2)_weak, and U(1)_em define interactions in the Standard Model; spontaneous symmetry breaking and the Higgs mechanism involve Lie group structure and representations. In condensed matter physics continuous symmetries and their breaking explain phenomena such as superconductivity and magnetism; groups like U(1) and SO(3) appear in effective field theories. Quantum information uses Lie groups for unitary gates (U(n), SU(2^n)) and continuous control in quantum computing proposals pursued by laboratories including IBM, Google and academic groups.
Compact simple Lie groups are classified by Dynkin diagrams into families A_n, B_n, C_n, D_n and exceptional types (E6, E7, E8, F4, G2). Physically prominent series include SU(n) (A_{n-1}), SO(n) (B_n, D_n), and Sp(n) (C_n). Exceptional groups occasionally appear in theoretical models (e.g., grand unified theories invoking E6 or E8), and infinite-dimensional Lie groups like loop groups and the Virasoro algebra play roles in conformal field theory and string theory. Low-dimensional examples—U(1), SU(2), SL(2,C), Heisenberg group—serve as testbeds for quantization, representation classification, and spectral analysis.
Analysis on Lie groups employs tools from topology and measure theory. A locally compact Lie group carries a unique (up to scale) left-invariant Haar measure used to define group convolutions, Fourier transforms, and path integral measures in quantum field theory. Techniques from differential topology (fiber bundles, principal bundles) and cohomology classify gauge field configurations and topological sectors (instantons, monopoles). Harmonic analysis on groups, including the use of characters and Plancherel theorems, underlies the decomposition of physical states and scattering amplitudes in both nonrelativistic and relativistic quantum theories.
Category:Quantum physics Category:Lie groups Category:Mathematical physics